Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model

Florin Avram, Dan Goreac, Juan Li and Xiaochi Wu
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Florin Avram: Laboratoire de Mathématiques Appliquées, Université de Pau, F-64012 Pau, France
Dan Goreac: School of Mathematics and Statistics, Shandong University (Weihai), Weihai 264209, China
Juan Li: School of Mathematics and Statistics, Shandong University (Weihai), Weihai 264209, China
Xiaochi Wu: School of Mathematics and Statistics, Shandong University (Weihai), Weihai 264209, China

Mathematics, 2021, vol. 9, issue 9, 1-27

Abstract: We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using the absolute continuity of super-solutions of a convenient Hamilton-Jacobi variational inequality. As a by-product, for exponential claims, we prove the optimality of bounded buffer capital injections ( ? a , 0 , b ) policies. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit a , and only pay dividends when the reserve reaches an upper barrier b . An exhaustive and explicit characterization of optimal couples buffer/barrier is given via comprehensive structure equations. The optimal buffer is shown never to be of de Finetti ( a = 0 ) or Shreve-Lehoczy-Gaver ( a = ? ) type. The study results in a dichotomy between cheap and expensive equity, based on the cost-of-borrowing parameter, thus providing a non-trivial generalization of the Lokka-Zervos phase-transition Løkka-Zervos (2008). In the first case, companies start paying dividends at the barrier b * = 0 , while in the second they must wait for reserves to build up to some (fully determined) b * > 0 before paying dividends.

Keywords: optimal dividends; capital injections; Cramér-Lundberg model; buffered reflection; scale functions; absolutely continuous supersolutions; Lokka-Zervos-type alternative (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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